Hadamard Tensors and Lower Bounds on Multiparty Communication Complexity

We develop a new method for estimating the discrepancy of tensors associated with multiparty communication problems in the ``Number on the Forehead\u27\u27 model of Chandra, Furst and Lipton. We define an analogue of the Hadamard property of matrices for tensors in multiple dimensions and show that any $k$-party communication problem represented by a Hadamard tensor must have $Omega(n/2^k)$ multiparty communication complexity. We also exhibit constructions of Hadamard tensors, giving $Omega(n/2^k)$ lower bounds on multiparty communication complexity for a new class of explicitly defined Boolean functions

Hadamard tensors and lower bounds on multiparty communication complexity

We develop a new method for estimating the discrepancy of tensors associated with multiparty communication problems in the “Number on the Forehead ” model of Chandra, Furst and Lipton. We define an analogue of the Hadamard property of matrices for tensors in multiple dimensions and show that any k-party communication problem represented by a Hadamard tensor must have Ω(n/2 k) multiparty communication complexity. We also exhibit constructions of Hadamard tensors, giving Ω(n/2 k) lower bounds on multiparty communication complexity for a new class of explicitly defined Boolean functions.